جستجوی مقالات مرتبط با کلیدواژه « laplacian energy » در نشریات گروه « ریاضی »

Let $G$ be a graph with a vertex weight $omega$ and the vertices $v_1,ldots,v_n$. The Laplacian matrix of $G$ with respect to $omega$ is defined as $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $mu_1,cdots,mu_n$ be eigenvalues of $L_omega(G)$. Then the Laplacian energy of $G$ with respect to $omega$ defined as $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. In this paper we consider several natural vertex weights of $G$ and obtain some inequalities between the ordinary and Laplacian energies of $G$ with corresponding vertex weights. Finally, we apply our results to the molecular graph of toroidal fullerenes (or achiral polyhex nanotorus).\[5mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.

In this paper, we investigate some of the graph energies of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$. Let $Z(R)$ be the set of zero-divisors of a commutative ring $R$ with non-zero identity and $Z^*(R)=Z(R)\setminus \{0\}$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set in $Z^*(R)$ and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$. We investigate some energies of $\Gamma(R)$ for the commutative rings $R\simeq \mathbb{Z}_{p^2}\times \mathbb{Z}_{q}$, $R\simeq \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}$ and $R\simeq \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}$ where $p, q$ the prime numbers.

In this paper, some graph parameters of the zero-divisor graph $\Gamma(R)$ of a finite commutative ring $R$ for $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $R\simeq \mathbb{Z}_p \times \mathbb{Z}_{2p}$ where $p>2$ a prime, are investigated. The graph $\Gamma(R)$ is a simple graph whose vertex set is the set of non-zero zero-divisors of a commutative ring $R$ with non-zero identity and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$.
In this paper, we study some of the topological indices such as graph energies, the Zagreb indices and the domination parameters of graphs $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{p^2} \big)$ and $\Gamma\big(\mathbb{Z}_p \times \mathbb{Z}_{2p}\big)$.

Let $G$ be a simple connected graph of order $n$ and size $m$. The matrix $L(G)=D(G)-A(G)$ is the Laplacian matrix of $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix, respectively. For the graph $G$, let $d_{1}\geq d_{2}\geq \cdots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}>\mu_{n}=0$ be the Laplacian eigenvalues. The Laplacian resolvent energy $RL(G)$ of a graph $G$ is defined as $RL(G)=\sum\limits_{i=1}^{n}\frac{1}{n+1-\mu_{i}}$. In this paper, we obtain an upper bound for the Laplacian resolvent energy $RL(G)$ in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy $RL(G)$ with each of the Laplacian-energy-Like invariant $LEL$, the Kirchhoff index $Kf$ and the Laplacian energy $LE$ of the graph.

‎Let $G$ be a finite non-abelian group with center $Z(G)$‎. ‎The non-commuting graph of $G$ is a simple undirected graph whose vertex set is $Gsetminus Z(G)$ and two vertices $x$ and $y$ are adjacent if and only if $xy ne yx$‎. ‎In this paper‎, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups‎..

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini and Panahbar [R. Sharafdini, H. Panahbar, Vertex weighted Laplacian graph energy and other topological indices. J. Math. Nanosci. 2016, 6, 57-65], in this paper we investigate the eccentricity version of Laplacian energy of a graph G.

For a simple connected graph G with n -vertices having Laplacian eigenvalues μ 1, μ 2, …, μ n−1, μ n =0, and signless Laplacian eigenvalues q 1, q 2, …, q n, the Laplacian-energy-like invariant(LEL) and the incidence energy (IE) of a graph G are respectively defined as LEL(G)=∑ n−1 i=1 μ i − − √ and IE(G)=∑ n i=1 q i √. In this paper, we obtain some sharp lower and upper bounds for the Laplacian-energy-like invariant and incidence energy of a graph.

For a simple digraph $G$ of order $n$ with vertex set ${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^-$ denote the out-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let $D^+(G)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and $D^-(G)=diag(d_1^-,d_2^-,ldots,d_n^-)$. In this paper we introduce $widetilde{SL}(G)=widetilde{D}(G)-S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $widetilde{D}(G)=D^+(G)-D^-(G)$ and $S(G)$ is the skew-adjacency matrix of $G$, and from which we define the skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms of all the eigenvalues of $widetilde{SL}(G)$. Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.